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On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis Chapman University Chapman University Digital Commons Computational and Data Sciences (PhD) Dissertations Dissertations and Theses Spring 5-2020 On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis Ismael L. Paiva Chapman University, [email protected] Follow this and additional works at: https://digitalcommons.chapman.edu/cads_dissertations Part of the Analysis Commons, and the Quantum Physics Commons Recommended Citation I. Paiva, "On quantum effects of vector potentials and generalizations of functional analysis," Ph.D. dissertation, Chapman University, Orange, CA, Year. https://doi.org/10.36837/chapman.000159 This Dissertation is brought to you for free and open access by the Dissertations and Theses at Chapman University Digital Commons. It has been accepted for inclusion in Computational and Data Sciences (PhD) Dissertations by an authorized administrator of Chapman University Digital Commons. For more information, please contact [email protected]. On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis A Dissertation by Ismael Lucas de Paiva Chapman University Orange, CA Schmid College of Science and Technology Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computational and Data Sciences May 2020 Committee in charge: Jeff Tollaksen, Ph.D., Chair Yakir Aharonov, Ph.D. Daniel Alpay, Ph.D. Justin Dressel, Ph.D. Augusto C. Lobo, Ph.D. Sandu Popescu, Ph.D. Daniele C. Struppa, Ph.D. pprov d. Dani 1 Alpa Ph.D. IL_/ Ju in Dr 1, Ph.D. ugusto C. Lobo Ph.D. Sandu Pope cu Ph.D. Daniel . Struppa Ph.D. May 2020 On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis Copyright © 2020 by Ismael Lucas de Paiva III ACKNOWLEDGMENTS Many people and organizations contributed directly or indirectly to my studies. I wish to pay gratitude to them. First, I would like to thank my parents, Geraldo and N´adia,for their continuous and unconditional support throughout different stages of my life. Also, I would like to thank the friendship from my siblings, Rafael and Miri~a.I appreciate the privilege of growing up knowing that my family would always be there for me. I would also like to thank all teachers and professors that contributed to my academic formation. I am truly grateful to all of them. To mention a few (in chronological order of contribution to my studies), I should begin by giving a special thanks to K´atiaVilela. She was one of my high school teachers and, besides all the effort put in her lectures, she also recommended me to participate in a Junior Scientific Initiation at the Federal University of Ouro Preto. It was then, studying with Fl´avioCassino, that I started making my mind about pursuing a career in physics. Following that, I was advised by Augusto Lobo during my undergraduate. I have no words good enough to thank him. His advising continued during my masters, this time with the participation of Maria Carolina Nemes, who is not with us anymore but had a big impact on me with her constant excitement about physics and her dedication to her students. Finished my masters, I started my applications to start my doctorate studies at Chapman University. If it were not the assistance from Jeff Tollaksen, one of my Ph.D. advisors, since the time I was still in Brazil trying to get a fellowship, I would not have gotten here. He has supported me throughout this entire time, always trying to make sure I was taken care of. Moreover, I am incommensurately grateful for all the generosity of Yakir Aharonov. I do not know how to thank him for his help on my applications for the fellowship to come to Chapman, his effort to guide me on my physics research, and his care. Furthermore, I would like to thank Daniel Alpay, who has advised me on my mathematical research. We started IV working during a period where I was thinking that I would not succeed in my doctorate studies. Thank you for all the time you dedicated to working with me. I also thank Daniele Struppa for the collaboration in our projects. Finally, I am in great debit with Mordecai (Cai) Waegell. I do not believe I would have successfully concluded the physics research that is presented in this dissertation if it was not our long debates and for his important contributions. Thank you very much! Besides the aforementioned names, I would like to also thank the physics professors at Chapman University. In particular, I thank Justin Dressel for our many discussions in his office. I learned a great deal with you. I am also grateful for the discussions and interactions with Matthew Leifer, Roman Buniy, and Amir Ahsan. I would also like to thank all my friends. In special, I thank Shiva Lotfallahzadeh Barzili. From getting at Chapman as international students up until now, we shared our journey through the doctorate studies. Thank you for your friendship. I am also grateful for my landlords, Lawrence and Lois Reed. They gave me support since my first weeks in the United States. They also inserted me in their family in such a way that, now, they are a second family to me. Besides them, I would also like to mention some of the friends who were important in my journey (in alphabetic order): Kyle Anderson, Lorenzo Catani, Dana Dacier, Sidy Danioko, Erik de Oliveira Martins, Pedro Dieguez, Cristhiano Duarte, Sam Ford, Eric Freda, Luis Pedro Garc´ıa-Pintos, Razieh Mohseninia, Robin Pendergraft, Cyril Rakovski, Viseth Sean, Arjun Tummala, Adrian Vajiac, Mihaela Vajiac, Kristen Whitney, Jianwei (Arnold) Zheng. This list of names is not intended to be complete, and I ask pardon from anyone who was not included in it. Moreover, I would like to thank all the funding agencies/institutions that supported me. In special, I would like to thank the Science without Borders program from the Brazilian government. Unfortunately, it does not exist anymore, but I would never have dreamed of V studying in a different country if it was not for this program. The fellowship that I got through this program was funded by the National Council for Scientific and Technological Development (CNPq), an agency that has supported me in different stages of my studies. I am grateful that I was studying at a time when so many opportunities were available to students like me. I also thank Chapman University for the teaching assistantship that was offered to me. Furthermore, I thank the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust, which provided some of the funds that were used in travels related to my research. Finally, I thank the members of the committee for the time and work put on reading this dissertation. In special, I would like to thank Sandu Popescu, who was not mentioned by name in these acknowledgments yet. I really value the participation of every committee member and their suggestions for future works. To all mentioned here, muit´ıssimoobrigado! VI LIST OF PUBLICATIONS D. Alpay, I. L. Paiva, and D. C. Struppa, Distribution spaces and a new construction of stochastic processes associated with the Grassmann algebra, J. Math. Phys. 60, 013508 (2019). I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Topological bound states for quantum charges, Phys. Rev. A 100, 040101(R) (2019). D. Alpay, I. L. Paiva, and D. C. Struppa, Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra, Integr. Equ. Op. Th. 91, 8 (2019). I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Magnetic forces in the absence of a classical magnetic field, Phys. Rev. A 101, 042111 (2020). D. Alpay, I. L. Paiva, and D. C. Struppa, A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics, Isr. J. Math. 236, 207 (2020). D. Alpay and I. L. Paiva, On the extension of positive definite kernels to topological algebras, arXiv:2003.05632 (2020). I. L. Paiva, Y. Aharonov, J. Tollaksen, and M. Waegell, Complex-valued vector poten- tial in a pre and post-selected system, (in preparation) (2020). VII ABSTRACT On Quantum Effects of Vector Potentials and Generalizations of Functional Analysis by Ismael Lucas de Paiva This is a dissertation in two parts. In the first one, the Aharonov-Bohm effect is investigated. It is shown that solenoids (or flux lines) can be seen as barriers for quantum charges. In particular, a charge can be trapped in a sector of a long cavity by two flux lines. Also, grids of flux lines can approximate the force associated with continuous two-dimensional distributions of magnetic fields. More, if it is assumed that the lines can be as close to each other as desirable, it is explained how the classical magnetic force can emerge from the Aharonov- Bohm effect. Continuing, the quantization of the source of the magnetic field, and not just of the degrees of freedom of the particle interacting with it, is considered. Special attention is given to the cases where the source has a relatively small spreading and is post-selected. As it will be discussed, in those cases, the weak value plays a role in the determination of the effective vector potential \experienced" by the particle. In the second part of this work, notions from functional analysis are extended to Banach algebras and completions of Grassmann algebras. A notion of analyticity is given to the functions of a single Banach algebra variable. This notion allows the introduction of holomorphic polynomials, power series, and rational functions. With that, the analogous of Hilbert spaces of power series are also considered. Finally, closures of Grassmann algebras with respect to the 1 and the 2-norms are explored.
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